Eigen Value

An eigenvalue of a n by n matrix A is a scalar c such that A*x = c*x holds for some nonzero vector x (where x is an n-tuple). See first: EigenVector, MatrixFactoring, MatrixAnalysis.

The magnitude of the EigenValue = (1 + interest rate).

The imaginary part of the EigenValue tells a "natural" frequency of cycles in the system.

One reason this is worth knowing about is that if the matrix represents somehow what a system that you are interested in does then the eigen values and vectors (eigen means "characteristic" in German, if I remember correctly) can be used to build a framework for describing that system and its actions that has many convenient features.

eigen means "own" and Eigenschaft means "characteristic.

NetPresentValue calculations use EigenValues.

Let's say you have two countries, Blueland (pop. 10 million) and Redland (60 million). Ignoring immigration for a second, Blueland's population grows by 5% a year, and Redland grows by 10% a year. But each year, only 8% of Bluelanders move to Redland, whereas 35% of Redlanders move to Blueland. Blueland's population should grow faster than Redland at first, but as you had fewer (relative) Redlanders, Blueland's overall growth rate would slow, and you'd expect the ratio of populations to eventually converge on some value. But what is that ratio? Also, how much would each country's population grow per year once that steady state was reached?

-- SteveHowell

The eigenvalue is the a multiplicative factor for the transformation. The matrix A is a linear transformation operator in n-dimension (i.e. A*x is a linear transformation for vector x, linear in the sense that A*(2x) == 2(A*x)). Consider the special case of n==2 (i.e. 2 dimension), and vector x = (x,y) representing positions in a plane (think a piece of paper). The effect of A is to move points on the plane from vector x=(x,y) to vector A*x=(x',y'). For some kind transformation (i.e. a subset of all possible A), you can find [eigenvalue c, eigenvector v] pairs with the special property that A*v=cv. There will be exactly n pairs of them. Geometrically, the set {v} represents the directions where the application of A will not change the orientation (hence A*v equals a number times v). In other words, the effect of A is to either "stretch" (if c>1) or "squeeze" (if c<1) or "reflect" (if c<0) along the direction v. So with the set of [c,v]'s, you can quick "see" what the effect of A is.

--OliverChung

Can you somehow factor a matrix into a simple combination of its EigenVectors?

See MatrixFactoring

I think the geometry analogy can be applied to the Blueland/Redland problem above. Each year, graph one point that represents the current state of the world--make the Blueland population be your x and Redland your y. The population transformation matrix can be found as follows:

 B1 = population of Blueland at time 1.
 B2 = population of Blueland one year later.

 R1 = population of  Redland at time 1.
 R2 = population of  Redland one year later.

 B2 = 1.05 * B1 - 0.08 * B1 + 0.35 * R1
 R2 = 1.10 * R1 + 0.08 * B1 - 0.35 * R1

 B2 = 0.97 * B1 + 0.35 * R1
 R2 = 0.08 * B1 + 0.75 * R1

 [B2]   [ 0.97   0.35 ]   [B1]
 [  ] = [             ] * [  ]
 [R2]   [ 0.08   0.75 ]   [R1]

 So the population transformation matrix is
 0.97 0.35
 0.08 0.75

When you plot the points for each year, you'll see that they eventually begin to stretch roughly along the same line. The line that they stretch about is determined by an eigenvector of the population transformation matrix.

-- SteveHowell, AnonymousDonor

After a long time, we have (approximately, but still needing more round-off): (Please correct any errors, especially in the definitions)

 [B2]   [ 0.97   0.35 ]   [B1]             [0.795]             
 [  ] = [             ] * [  ] = 1.06025 * [     ] * (B1 + R1)
 [R2] = [ 0.08   0.75 ]   [R1]             [0.205]

 This eigenvector is:
 0.795
 0.205

 This eigenvalue is:
 1.06025

There will eventually be about 4 times as many blues as reds.

The populations will eventually both grow about 6 percent per year.

There is another eigenvector / eigenvalue pair as well, but it is an impractical answer. This answer should also be rounded off:

 [B2]   [ 0.97   0.35 ]   [B1]             [ 8.805]             
 [  ] = [             ] * [  ] = 0.65975 * [      ] * (B1 + R1)
 [R2] = [ 0.08   0.75 ]   [R1]             [-7.805]

 This eigenvector is:
  8.805
 -7.805

 This eigenvalue is:
  0.65975

There are a negative number of reds in this answer.

CategoryMath, NetPresentValue